q
-distributions in complex systems: a brief review
S. Picoli Jr., R. S. Mendes, L. C. Malacarne and R. P. B. Santos
Departamento de F´ısica and National Institute of Science and Technology for Complex Systems, Universidade Estadual de Maring´a, Avenida Colombo 5790,
87020-900 Maring´a, PR, Brazil
(Received on 26 March, 2009)
The nonextensive statistical mechanics proposed by Tsallis is today an intense and growing research field. Probability distributions which emerges from the nonextensive formalism (q-distributions) have been applied to an impressive variety of problems. In particular, the role ofq-distributions in the interdisciplinary field of complex systems has been expanding. Here, we make a brief review ofq-exponential,q-Gaussian andq-Weibull distributions focusing some of their basic properties and recent applications. The richness of systems analyzed may indicate future directions in this field.
Keywords:q-exponential,q-Gaussian,q-Weibull, Nonextensive statistics
1. INTRODUCTION
Common characteristics of complex systems include long-range correlations, multifractality and non-Gaussian distribu-tions with asymptotic power law behavior. Typically, such systems are not well described by approaches based on the usual statistical mechanics. In this scenario, a new formalism capable of providing a better description of complex systems is welcome. This is the case of the generalized (nonextensive) statistical mechanics proposed by Tsallis - nowadays, an in-tense and growing research field[1–4].
Concepts related with nonextensive statistical mechanics have found applications in a variety of disciplines including physics, chemistry, biology, mathematics, geography, eco-nomics, medicine, informatics, linguistics among others[5– 7]. Probability distributions which emerge from the nonex-tensive formalism - also calledq-distributions - have been ap-plied to an impressive variety of problems in diverse research areas including the interdisciplinary field of complex systems. In the present work we focus onq-exponential,q-Gaussian andq-Weibull distributions. We summarized some of their basic properties and provide useful references of recent ap-plications. The richness of systems analyzed may indicate future directions in this research line.
2. q-EXPONENTIAL DISTRIBUTION
The q-exponential distribution is given by the probability density function (pdf)
pqe(x) =p0
1−(1−q) x x0
1/(1−q)
(1)
for 1−(1−q)x/x0≥0. Ifp0= (2−q)/x0, eq. (1) is
normal-ized.
In the limit q→1, eq. (1) recovers the usual exponen-tial distribution in the same way in which theq-exponential function, defined ase−qx≡[1−(1−q)x]1/(1−q), recovers
ex-ponential function in the limitq→1 (e−1x≡e−x). Ifq<1, eq. (1) has a finite value for any finite real value of x since, by definition,pqe(x) =0 for 1−(1−q)x/a<0. Ifq>1, eq.
(1) exhibits power law asymptotic behavior,
pqe(x)∼x−1/(q−1). (2)
0 1 2 3 4
0,0 0,5 1,0
q=2,5 q=1,5 q=1,0 q=0,5
pqe
(x
)
x
(a)
0,1 1 10 100
10-3
10-1
q=2,5 q=1,5 q=1,0 q=0,5 pqe
(x
)
x
(b)
0 1 2 3
-2 -1 0
x0=2.5 x0=1.5 x
0=1.0 x0=0.5
lnq pqe
(x
)
x (c)
FIG. 1:q-exponential distribution. a) Plot ofpqe(x)versusx, with
p0=x0=1 and typical values ofq. b) Log-log plot of the curves in
a). c) lnqpqe(x)versusxforp0=1 and typical values ofx0.
Note also thatpqe(x)≃1+xfor smallx, independently of the
q value. Figures 1a and 1b showpqe(x)versusxfor typical
values ofq.
100 102 104 106 101
103
R
(P
)
P USA
(a)
0,0 5,0x106
1,0x107
0 1
lnq
’
R
(P
)
P USA
(b)
103
104
105
106
107
10-1
101 103
R
(P
)
P
(c)
Brazil
0,0 5,0x106
1,0x107
0,0 0,5 1,0 1,5
lnq
’
R
(P
)
P (d)
Brazil
FIG. 2:Population of cities.a) Empirical cdfR(P), wherePis the population of USA cities. The solid line is aq-exponential, given by eq. (3), withq′=1.7 (q≃1.4),x′0=21,250 andc′=2,919. b) lnq′R(P)versusP, withq′=1.7, for the same data shown in (a). The solid line
is a linear fit to the data. c) Empirical cdfR(P), wherePis the population of Brazilian cities. The solid line is aq-exponential, given by eq. (3), withq′=1.7 (q≃1.4),x′0=7,073 andc′=6,968. d) lnq′R(P)versusP, withq′=1.7, for the same data shown in (c). The solid line is
a linear fit to the data.
103
104
105
106
107
101
103
R
(S
)
S
(a)
USA
0 1x107
2x107
0,4 0,8 1,2 1,6
lnq
’
R
(S
)
S
(b)
USA
103
104
105
106
107
101
103
R
(S
)
S
(c)
UK
0 1x106
2x106
3x106
0,4 0,8 1,2 1,6
lnq
’
R
(S
)
S
(d)
UK
FIG. 3:Circulation of magazines. a) Empirical cdfR(S), whereSis the circulation of 570 USA magazines in 2004. The solid line is a
q-exponential, given by eq. (3), withq′=1.65 (q≃1.4),x′0=255,204 andc′=594. b) lnq′R(S)versusS, withq′=1.65, for the same data
shown in (a). The solid line is a linear fit to the data. c) Empirical cdfR(S), whereSis the circulation of 727 UK magazines in 2005. The solid line is aq-exponential, given by eq. (3), withq′=1.65 (q≃1.4),x′0=37,493 andc′=860. b) lnq′R(S)versusS, withq′=1.65, for
the same data shown in (c). The solid line is a linear fit to the data.
The cumulative distribution function (cdf) associated to eq. (1) is given by
Rqe(x) =
Z ∞
x
pqe(y)dy
= p′0
1−(1−q′) x x′0
1/(1−q′)
, (3)
defined forq<2, withq′=1/(2−q),x′0=x0/(2−q)and
p′0=p0x0/(2−q). Observe thatRqe(x)andpqe(x)exhibit the
same mathematical form.
It is possible to visualize q-exponential distributions as straight lines in graphs with appropriate scales. Applying the q-logarithm function, defined as lnqx≡[x(1−q)−1]/(1−q),
with ln1x≡ln(x), in both sides of eq. (1), we have
lnqpqe(x) =lnqp0−[1+ (1−q)lnqp0]
x x0
A similar result holds forRqe(x). Figure 1c shows lnqpqe(x)
versusxfor typical values ofx0.
The q-exponential function given by eq. (1) has been employed in a growing number of theoretical and empiri-cal works on a large variety of themes. Examples include scale-free networks[10–14], dynamical systems[15–27], al-gebraic structures[28–31] among other topics in statistical physcics[32–36].
As specific examples of q-exponential distributions in complex systems, let us consider results on population of cities[37] and circulation of magazines[38]. Figure 2 shows the cumulative distribution of the population of cities in the USA and Brazil. Figure 3 shows the cumulative distribution of circulation of magazines in the USA and UK. In both cases - population of cities and circulation of magazines - the em-pirical data are consistent with aq-exponential distribution, withq≃1.4.
q-exponential distributions have also been applied in the empirical study of stock markets[39–42], DNA sequences[43], family names[44], human behavior[45–47], geomagnetic records[48, 49], train delays[50], reaction kinetics[51], air networks[52], hydrological phenomena[53], fossil register[54], basketball[55], earthquakes[56–58], world track records[59], voting processes[60], internet[61], individual success[62], citations of scientific papers[63, 64], football[65], linguistics[66, 67] and solar neutrinos[68, 69].
3. q-GAUSSIAN DISTRIBUTION
Theq-Gaussian distribution is specified by the pdf
pqg(x) =p0
"
1−(1−q) x
x0
2#1/(1−q)
, (5)
for 1−(1−q)(x/x0)2≥0 and pqg(x) =0 otherwise. It is
normalized ifp0= (2/x0)
p
(q−1)/π Γ[1/(q−1)]/Γ[1/(q− 1)−1/2]. In addition, eq. (5) presents unit variance ifx20= 5−3q, withq<5/3.
In the limitq→1, eq. (5) recovers the usual Gaussian dis-tribution, soq6=1 indicates a departure from Gaussian statis-tics. Forq>1, the tails ofq-Gaussian decrease as power laws,
pqg(|x|)∼ |x|−2/(q−1). (6)
Figures 4a and 4b showpqg(x)for typical values ofq.
Applying theq-logarithm function in both sides of eq. (5), we have
lnqpqg(x) =lnqp0−[1+ (1−q)lnqp0]
x
x0
2
. (7)
Figure 4c shows lnqpqg(x)versusx2for typical values ofx0.
Recent works have been focused on the study of mathe-matical properties of q-Gaussian functions[70–78], includ-ing methods for generatinclud-ing random numbers which fol-low q-Gaussian distributions[79, 80]. q-Gaussians have been employed in the study of a wide range of themes in-cluding probabilistic models[81, 82], stellar plasmas[83], porous-medium equation[84], Bose-condensed gases[85–87],
-4 -2 0 2 4
0 2 4
q=1.7 q=1.5 q=1.3 q=1.0
pqg
(x
)
x
(a)
-8 -4 0 4 8
0,01 0,1 1 10
q=1.7 q=1.5 q=1.3 q=1.0
pqg
(x
)
x
(b)
0 4 8
-6 -3 0
x0=1.7 x0=1.5 x0=1.3 x0=1.0 lnq
pq
g
(x
)
x2
(c)
FIG. 4: q-Gaussian distribution. a) Plot ofpqg(x)versusx, with
p0=x0=1, for typical values ofq. Some curves were vertically
shifted for a better visualization. b) The same curves shown in a), but for mono-log scale. Some curves were also shifted . c) lnqpqg(x)
versusx2forp0=1 and typical values ofx0.
dynamical systems[88–90], polymeric networks[91], small-world networks[92], fingering processes[93], processes with stochastic volatility[94, 95] and nonlinear diffusion[96, 97].
In order to illustrate a recent application ofq-Gaussian dis-tributions in complex systems, we mention here results on the dynamics of earthquakes[98]. Figure 5 shows the distribution of energy differences between successive earthquakes at the San Andreas Fault. The empirical data is consistent with a q-Gaussian distribution, withq=1.75.
Other recent applications of q-Gaussian distribution in-clude stock markets[99–107], DNA molecules[108], the solar wind[109–111], galaxies[112], optical lattices[113], cellular aggregates[114] and the atmosphere[115].
4. q-WEIBULL DISTRIBUTION
Theq-Weibull distribution is given by the pdf
pqw(x) =p0
rxr−1
xr0
1−(1−q) x
x0
r1/(1−q)
, (8)
for 1−(1−q)(x/x0)r≥0 andpqw(x) =0 otherwise. Eq. (8)
is normalized ifp0=2−q.
re--40 -20 0 20 40 10-5
10-3 10-1
P
(Z
)
Z
(a)
0 10 20 30
-400 -200 0
lnq
p
(Z
)
Z2
(b)
FIG. 5:Earthquakes. a) Empirical pdfP(Z), whereZ=E(t+1)−E(t)is the energy difference between successive earthquakes at the San Andreas Fault in the period 1966-2006. The solid line is aq-Gaussian, given by eq. (5), withq=1.75,x0=0.25 andp0=1.63. b) lnqP(Z)
versusZ2, withq=1.75, for small values ofZ. The solid line is a linear fit to the data.
0 2 4
0,0 0,4 0,8
r=1.0; q=1.5 r=1.5; q=1.0 r=1.5; q=1.5
pqw
(x
)
x
(a)
0,1 1 10
10-2
10-1
100
r=2.0
q=2.0 q=1.7 q=1.4 q=1.0
pqw
(x
)
x
(b)
0,1 1 10
10-2
10-1
100
q=1.5
r=2.5 r=2.0 r=1.5 r=1.0
pqw
(x
)
x
(c)
0 4 8
-6 -3 0
x0=1.7 x0=1.5 x0=1.3 x
0=1.0
lnq
’
R
q
w
(x
)
xr
(d)
FIG. 6:q-Weibull distribution. a) Plot ofpqw(x)versusx, withp0=x0=1, and typical values ofqandr. b) Log-log plot ofpqw(x)versus
x, withp0=x0=1,r=2 and typical values ofq. c) a) Log-log plot ofpqw(x)versusx, withp0=x0=1,q=1.5 and typical values ofr. d)
lnq′Rqw(x)versusxrforp′0=1 and typical values ofx0.
spectively. Ifq<1,pqw(x)has a finite limit sincepqw(x) =0
for 1−(1−q)(x/x0)r<0. Ifq>1,pqw(x)exhibits power law
behavior both for small and large values ofx. More specifi-cally,
pqw(x)∼x−ξ, (9)
withξ= (1−r)for smallxandξ=r[(2−q)/(q−1)] +1 for largex. Figures 6a, 6b and 6c show pqw(x)versusxfor
typical values ofqandr.
The cdf associated topqw(x)is given by
Rqw(x) =p′0
1−(1−q′)
x x′0
r1/(1−q′)
, (10)
withq′=1/(2−q),(x′0)r=xr
0/(2−q)andp0′ =p0/(2−q).
Applying theq-logarithm function in both sides of the cdf Rqw, we have
lnq′Rqw(x) =lnq′p′0−[1+ (1−q)lnq′p′0] x
x0
r
. (11)
Figure 6c shows lnq′RqW(x)versusxrfor typical values ofx0.
If pqw(x)is normalized (p0=2−q), Eq. (11) reduces to
lnq′Rqw(x) =−(x/x0)r. In this case,
ln[−lnq′(Rqw(x))] =rlnx−rlnx0. (12)
As specific example ofq-Weibull distribution in complex systems, we now consider results on citations in scientific journals[116]. Figure 7 shows the distribution of the impact factor of scientific journals in comparison with aq-Weibull curve. The empirical data is consistent with aq-Weibull dis-tribution, withq=1.45 andr=1.50.
0,01 0,1 1 10 100 10-3
10-1
p
(F
)
F
(a)
0 100 200 300
0,6 0,9 1,2
lnq
’
R
(F
)
Fr
(b)
FIG. 7: Citations in scientific journals. a) Empirical pdf p(F), whereFis the 2004 impact factor for 5912 scientific journals. The solid line is aq-Weibull distribution, given by eq. (8), withr=1.5,
q=1.45,x0=0.74 andp0=0.58. b) lnq′R(F)versusFr, with
q′=1.82 (q=1.45) andr=1.5. The solid line is a linear fit to the data.
5. BASIS FORq-DISTRIBUTIONS
¿From the viewpoint of the principle of the maximum en-tropy, someqdistributions optimizes generalized entropies -more general entropic measures than the standard Boltzmann-Gibbs entropy. A striking example is theq-entropy proposed by Tsallis[1]
Sq=
1−∑Wi=1p q i
q−1 , (13)
whereW is the total number of microstates of the system,pi
are the occupation probabilities andqis a real parameter. The standard Boltzmann-Gibbs entropy is recovered in the limit q→1.
The maximization ofSqsubject to specific constraints
gen-erates occupation probabilities following aq-exponential dis-tribution. Theq-exponential optimizes other generalized tropic measures such as the Renyi and normalized Tsallis en-tropies. However, only Tsallis entropy can provide an appro-priate basis for theq-exponential distribution since it presents several properties essential for an entropy[122, 123]. Chang-ing the constraints, the maximization ofSqalso generates
oc-cupation probabilities following aq-Gaussian distribution. Formally, q-distributions can arise when the exponen-tial function of the original distribution is replaced by a
q-exponential function. For example, this basic procedure ap-plied in standard exponential, Gaussian and Weibull distri-butions leads to q-exponential, q-Gaussian and q-Weibull, respectively[55]. This viewpoint suggests the consideration of other q-distributions which could be obtained by simply replacing its exponential function by aq-exponential one.
q-distributions can also emerge from compound distributions[124]
pq(x) =
Z ∞
0
p(x,λ)f(λ)dλ, (14)
where f(λ) is a Gamma function. For example, if p(x,λ) is a Weibull distribution, pq(x) is given by a q-Weibull
distribution[120]. Naturally, other forms for f(λ) may be considered to obtain alternative distributions. In a physical context, this scenario has been explored with success in su-perstatistics where nonequilibrium situations with local fluc-tuations of the environment are taken into account[125–127]. The generalized distributions considered here can also be obtained from the following ordinary differential equation:
dy dx=ρy
q.
(15)
In fact, if ρ is constant, the solution of eq. (15) is a q-exponential; if ρ ∝x, the solution is a q-Gaussian. If y is the cdf and ρ ∝xr, we have a q-Weibull. By consider-ing further terms in eq. (15), other q-distributions can be obtained[128]. q-distributions can also emerge in other con-texts. For instance, q-Gaussian arises from the non-linear diffusion (porous media) equation[84] and from a general-ization of the central limit theorem[3]. Another example is theq-lognormal distribution which emerges from generalized cascades[28].
6. CONCLUSION
The present work presents a brief overview of recent appli-cations of someq-distributions largely used in the context of Tsallis statistics. It illustrates howq-exponential,q-Gaussian andq-Weibull distributions have been applied in the study of a wide variety of systems in several fields.
The success of q-distributions in describing diverse sys-tems is in part due to its ability of exhibit heavy-tails and model power law phenomena - a typical characteristic of complex systems. The positive and exciting results obtained with q-distributions also indicate possible applications of Tsallis nonextensive statistical mechanics. Naturally, further work may be necessary to explore possible relations between the analyzed systems and the present theory.
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![Figure 4c shows ln q p qg (x) versus x 2 for typical values of x 0 . Recent works have been focused on the study of mathe-matical properties of q-Gaussian functions[70–78], includ-ing methods for generatinclud-ing random numbers which fol-low q-Gaussian](https://thumb-eu.123doks.com/thumbv2/123dok_br/18983351.457898/3.977.580.770.62.535/figure-typical-recent-properties-gaussian-functions-generatinclud-gaussian.webp)
